#8285503

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#59675	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,10)(3,8,11,6)(4,5,9,15)(12,13,16,14)(17,18)(19,28)(20,23)(21,31)(22,26)(24,27)(25,30)(29,32)(33,34,39,40)(35,42,37,41)(36,45,46,38)(43,48,47,44);(1,3,7,11)(2,8,10,6)(4,12,9,16)(5,13,15,14)(17,19)(18,25)(20,29)(21,26)(22,23)(24,30)(27,28)(31,32)(34,40)(36,46)(41,42)(43,47);(3,11)(6,8)(12,16)(13,14)(17,20,24,21)(18,23,27,31)(19,29,30,26)(22,28,32,25)(33,35,39,37)(34,41,40,42)(36,43,46,47)(38,48,45,44);(1,4)(2,5)(3,12)(6,14)(7,9)(8,13)(10,15)(11,16)(17,21,24,20)(18,23,27,31)(19,26,30,29)(22,28,32,25)(33,36)(34,38)(35,43)(37,47)(39,46)(40,45)(41,48)(42,44);(1,5)(2,9)(3,13)(4,10)(6,12)(7,15)(8,16)(11,14)(17,22)(18,26)(19,23)(20,28)(21,25)(24,32)(27,29)(30,31)(33,37,39,35)(34,42,40,41)(36,43,46,47)(38,48,45,44);(1,6)(2,3)(4,14)(5,12)(7,8)(9,13)(10,11)(15,16)(17,23)(18,20)(19,22)(21,27)(24,31)(25,29)(26,28)(30,32)(33,38)(34,36)(35,44)(37,48)(39,45)(40,46)(41,47)(42,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8192699);